**3. FOHS for linear advection: diffusion equation**

The two-dimension linear advection–diffusion equation is written as

$$\frac{\partial \phi}{\partial t} + a \frac{\partial \phi}{\partial \mathbf{x}} + b \frac{\partial \phi}{\partial \mathbf{y}} = \nu \left( \frac{\partial^2 \phi}{\partial \mathbf{x}^2} + \frac{\partial^2 \phi}{\partial \mathbf{y}^2} \right) + f(\mathbf{x}, \mathbf{y}), \tag{9}$$

where *ϕ* is a scalar function, which can be referred to as the primary solution variable, or the velocity potential. (*a*, *b*) is a constant advection vector, *ν* is a positive diffusion coefficient, and *f* (*x*, *y*) is the source term. The first-order derivatives of *ϕ* are introduced as additional variables

$$v\_{\mathfrak{x}} = \frac{\partial \phi}{\partial \mathfrak{x}}, \ v\_{\mathfrak{y}} = \frac{\partial \phi}{\partial \mathfrak{y}}.\tag{10}$$

By adding pseudo time (*τ*) derivatives with respect to both *ϕ* and additional variables, the following FOHS can be formulated.

*Hyperbolic Navier-Stokes with Reconstructed Discontinuous Galerkin Method DOI: http://dx.doi.org/10.5772/intechopen.109605*

$$\begin{cases} \frac{\partial \phi}{\partial \tau} + \frac{\partial \phi}{\partial t} + a \frac{\partial \phi}{\partial \mathbf{x}} + b \frac{\partial \phi}{\partial \mathbf{y}} = \nu \left( \frac{\partial v\_{\mathbf{x}}}{\partial \mathbf{x}} + \frac{\partial v\_{\mathbf{y}}}{\partial \mathbf{y}} \right) + f(\mathbf{x}, \mathbf{y}), \\\frac{\partial v\_{\mathbf{x}}}{\partial \tau} = \frac{1}{T\_{r}} \left( \frac{\partial \phi}{\partial \mathbf{x}} - v\_{\mathbf{x}} \right), \\\frac{\partial v\_{\mathbf{y}}}{\partial \tau} = \frac{1}{T\_{r}} \left( \frac{\partial \phi}{\partial \mathbf{y}} - v\_{\mathbf{y}} \right), \end{cases} \tag{11}$$

where *t* and *τ* are the physical time and the pseudo-time, respectively. If the steady solution in pseudo-time is obtained, then the extra variables would relax to the first-order derivatives of *ϕ*. *Tr* is the relaxation time, which is a free parameter. System (Eq. (11)) is equivalent (Eq. (9)) in the pseudo-steady state for any nonzero *Tr*, but *Tr* needs to be positive for the system to be hyperbolic. The FOHS can be written in vector form as

$$\frac{\partial \mathbf{U}}{\partial \boldsymbol{\pi}} + \mathbf{T} \frac{\partial \mathbf{U}}{\partial t} + \frac{\partial \mathbf{F}\_x}{\partial \boldsymbol{\kappa}} + \frac{\partial \mathbf{F}\_y}{\partial \boldsymbol{\chi}} = \mathbf{S},\tag{12}$$

where

$$\mathbf{U} = \begin{pmatrix} \phi \\ v\_{\times} \\ v\_{\times} \\ v\_{\times} \end{pmatrix}, \quad \mathbf{T} = \begin{pmatrix} \mathbf{1} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} \end{pmatrix}, \tag{13}$$

$$\mathbf{F}\_x = \begin{pmatrix} a\phi - \nu v\_x \\ -\phi/T\_r \\ 0 \end{pmatrix}, \quad \mathbf{F}\_y = \begin{pmatrix} b\phi - \nu v\_y \\ 0 \\ -\phi/T\_r \end{pmatrix}, \quad \mathbf{S} = \begin{pmatrix} f(x,y) \\ -v\_x/T\_r \\ -v\_y/T\_r \end{pmatrix}. \tag{14}$$

To simplify the mathematics, the advection term and the diffusive term are considered separately.

$$\mathbf{F}\_{\mathbf{x}} = \mathbf{F}\_{\mathbf{x}}^{d} + \mathbf{F}\_{\mathbf{x}}^{d} = \begin{pmatrix} a\phi \\ \mathbf{0} \\ \mathbf{0} \end{pmatrix} + \begin{pmatrix} -\nu v\_{\mathbf{x}} \\ -\phi/T\_{r} \\ \mathbf{0} \end{pmatrix},\tag{15}$$

$$\mathbf{F}\_{\mathcal{Y}} = \mathbf{F}\_{\mathcal{Y}}^{d} + \mathbf{F}\_{\mathcal{Y}}^{d} = \begin{pmatrix} b\phi \\ \mathbf{0} \\ \mathbf{0} \\ \mathbf{0} \end{pmatrix} + \begin{pmatrix} -\nu v\_{\mathcal{Y}} \\ \mathbf{0} \\ -\phi/T\_{r} \end{pmatrix}. \tag{16}$$

Consider the Jacobian of the flux projected along **n** ¼ *nx*, *ny* � �,

$$\mathbf{A}\_{n} = \mathbf{A}\_{n}^{a} + \mathbf{A}\_{n}^{d} = \frac{\partial \mathbf{F}\_{\mathbf{x}}}{\partial \mathbf{U}} n\_{\mathbf{x}} + \frac{\partial \mathbf{F}\_{\mathbf{y}}}{\partial \mathbf{U}} n\_{\mathbf{y}},\tag{17}$$

where **A***<sup>a</sup> <sup>n</sup>* and **A***<sup>d</sup> <sup>n</sup>* are the advective and diffusive Jacobians, respectively.

$$\mathbf{A}\_{n}^{a} = \frac{\partial \mathbf{F}\_{\times}^{a}}{\partial \mathbf{U}} \boldsymbol{n}\_{\times} + \frac{\partial \mathbf{F}\_{\times}^{a}}{\partial \mathbf{U}} \boldsymbol{n}\_{\times} = \begin{pmatrix} a\_{n} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} \end{pmatrix},\tag{18}$$

$$\mathbf{A}\_{\boldsymbol{n}}^{d} = \frac{\partial \mathbf{F}\_{\boldsymbol{\chi}}^{d}}{\partial \mathbf{U}} \boldsymbol{n}\_{\boldsymbol{\chi}} + \frac{\partial \mathbf{F}\_{\boldsymbol{\chi}}^{d}}{\partial \mathbf{U}} \boldsymbol{n}\_{\boldsymbol{\chi}} = \begin{pmatrix} \mathbf{0} & -\nu \boldsymbol{n}\_{\boldsymbol{\chi}} & -\nu \boldsymbol{n}\_{\boldsymbol{\chi}} \\ -\boldsymbol{n}\_{\boldsymbol{\chi}}/T\_{r} & \mathbf{0} & \mathbf{0} \\ -\boldsymbol{n}\_{\boldsymbol{\chi}}/T\_{r} & \mathbf{0} & \mathbf{0} \end{pmatrix},\tag{19}$$

where

$$a\_n = a n\_x + b n\_y. \tag{20}$$

The only nonzero eigenvalue of the advective Jacobian is *an*, while the eigenvalues of the diffusive Jacobian are

$$
\lambda\_1 = \sqrt{\frac{\nu}{T\_r}}, \quad \lambda\_2 = -\sqrt{\frac{\nu}{T\_r}}, \quad \lambda\_3 = 0. \tag{21}
$$

One can see that the diffusion progress can be regarded as a wave propagating isotropically according to the first two nonzero eigenvalues. As for the third eigenvalue, it indicates the inconsistency damping mode [18]. Clearly, the steady solution is independent of the free parameter, relaxation time *Tr*. Therefore, *Tr* can be defined solely to accelerate the convergence. For simplicity, *Tr* is taken as

$$T\_r = \frac{L\_r^2}{\nu}, \quad L\_r = \frac{1}{\max\left(Re, 2\pi\right)}, \quad Re = \frac{\sqrt{a^2 + b^2}}{\nu}. \tag{22}$$

One can see that only first-order operators occur in (Eq. (12)). FV formulation (Eq. (2)), DG formulation (Eq. (5)), or rDG formulation (Eq. (7)) can be used to integrate the first-order spatial operators.
